40 XII TOPOLOGY AND TOPOLOGICAL ALGEBRA From the definition of d, it follows immediately that d is left-invariant (because g is), satisfies the triangle inequality, is symmetric and positive. Moreover, the right-hand inequality in (12.9.1.2) is obvious, and shows that d(xy x) = 0 for all x e G. Hence dis & pseudo-distance on G. To prove the left- hand inequality of (12.9.1.2) we shall show by induction on p that, for each finite sequence (z£)0^i^p of p 4- 1 points in G such that z0 = x and zp = y, we have (12.9.13) I W W > iflfo JO- £ = 0 P- 1 This inequality is obvious if /? = !. Let us write a = 2_,g(zi9 zt+i)- The i = 0 inequality (12.9.1.3) is true if a ^ %, because y(x9y)^l. If a = 0, then 7. = z.+1 for 0 ^ f :g/> — 1, hence ;c = y and so (12.9.1.3) is trivially satis- fied. So suppose that 0 < a < ^, and let /z be the greatest index such that £ 0(zi9 zi+1) £ ia. Then we have % g(zt,zi+1) > -Jot, so that By the inductive hypothesis, we have g(x, zh) ^ a and g(zh + i, y) ^ a; on the other hand, it is clear that g(zh,zh+i) ^ a. Let k be the smallest integer >0 such that 2~* ^ a. Then A: ^ 2, and the elements x~1zh , z^lzh+1, z^^y are all in Vt , by virtue of the definition of g. Hence x"1y e V% c= Vfc^.1? which implies that ^(x, j) ^ 21"* ^ 2a5 and proves (12.9.13). Hence (12.9.1.2) is established, and shows first of all that dis a distance on G. Also, if r > 0, the ball &'(e; r) (with respect to d) contains V* for all indices such that 2~"k<r, and conversely each Vk contains the ball B'(e; 2~*~~1). Since dis left-invariant, it now follows from (12.8.3(i)) and (12.2.1) that d defines the topology of G. In general it is not possible to define the topology of G by means of a distance which is both left- and right-invariant (Problem 1 and Section 14.3, Problem 11). (12.9.2) Let G be a metrizable group. A necessary and sufficient condition for a sequence (xj in G to be a Cauchy sequence with respect to a left-invariant distance defining the topology ofG is that, for each neighborhood V ofe, there exists an integer n0 such that x~lxm e Vfor all m ^ «0 and n^.n0. For if d is a left-invariant distance, we have d(xn,*^ = d(e,x~lxm\ and the proposition follows from the fact that d defines the topology of G.duct of the topological groups G« ,